Properties

Label 578a
Number of curves 4
Conductor 578
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("578.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 578a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
578.a4 578a1 [1, 1, 1, -873, 5783] [2] 576 \(\Gamma_0(N)\)-optimal
578.a3 578a2 [1, 1, 1, -12433, 528295] [2] 1152  
578.a2 578a3 [1, 1, 1, -29773, -1989473] [2] 1728  
578.a1 578a4 [1, 1, 1, -32663, -1583717] [2] 3456  

Rank

sage: E.rank()
 

The elliptic curves in class 578a have rank \(0\).

Modular form 578.2.a.a

sage: E.q_eigenform(10)
 
\( q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + 4q^{7} + q^{8} + q^{9} - 6q^{11} + 2q^{12} + 2q^{13} + 4q^{14} + q^{16} + q^{18} - 4q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.